I like the look—more consistent with other stringed instruments. Flat paddles to me look more like toys.
The tuners all tighten the strings in the same turning direction, and turning the pegs is more ergonomic, less awkward.
The break angle across the nut is increased—similar tonal advantages as at the saddle (though, as far as I can work out, this would only affect strings played open, and I don't see the sense of having them sound different from stopped strings—rather, that strikes me as a disadvantage).
The buttons point back, as with friction tuners, for a somewhat sleek look—at least, from the front. (I often hear people say they prefer friction tuners for this reason, ignoring that, to the player, friction tuner pegs actually "stick out" more than standard geared tuners. But maybe they're more concerned with the visual impression they make on others—or spend more time looking at their ukes than playing them.)
With a slotted headstock, an ornate logo or headstock design is unnecessary. I'm not a fan of the kind of glitz often added to paddle headstocks.
Tuners for slotted headstocks are usually joined by a plate, often embossed with a design reminiscent of filigree—dressy without lapsing into gaudy.
Slotted headstock tuners tend to be of good quality, and are always open-geared and adjustable (at least, I haven't yet encountered an exception). There's no way to adjust a sealed tuner or UPT, apart from tightening the button, if you can even do that. Of course, many open-geared standard tuners are adjustable, and it costs less to replace them, should the need arise.
If there is added weight (remember, the slots take away some mass that the thickness adds), the "imbalance" may not matter if you play routinely with a strap, as I do.
A heavier headstock is said to aid the sound in some way: resonance? tone? sustain? I forget.
A slotted headstock carries more cachet as far as general public perception, partly because you seldom see cheap ukes with them—they cost more to make.
I've seen some cool slot designs—more striking and stylish than varying the paddle shape. The beveled bottom of a slot also shows off a layered headstock more effectively.